Will a neural network always predict the correct label if it sees the exact same input during training and testing?
No, Neural Networks do not have such a guarantee. In fact, I don't believe any kind of classifier in the entire field of Machine Learning has such a guarantee, though some may be slipping my mind...

For an easy counterexample, consider what happens if you have two instances with precisely identical inputs, but different output labels. If your classifier is deterministic (in the sense that there is no stochasticity in the procedure going from input to output after training), which a Neural Network is (unless, for example, you mess up a Dropout implementation and accidentally also apply dropout after training), it cannot possibly generate the correct output for both of those instances, even if they were presented as examples thousands of times during training.

Of course the above is an extreme example, but similar intuition applies to more realistic cases. There can be cases where getting the correct prediction on one instance would reduce the quality of predictions on many other instances if they have somewhat similar inputs. Normally, the training procedure would then prefer getting better predictions on the larger number of instances, and settle for failure on another instance.